Calculate the thickness of the coating using the Shewhart map. An example of constructing a Shewhart control chart in Excel. Shewhart control cards

01.03.2024 -

An example of constructing a Shewhart control chart in Excel

Shewhart control charts –one of the quality management tools. Used to monitor the progress of the process. As long as the values remain within the control limits, no intervention is required. Processstatistically controlled. If values are outside the control limits, management intervention is necessary to identify the causes of deviations.

Let's look at an example of constructing a control chart in Excel as part of accounts receivable management (for clarity, open the file Excel).

The source data contains information on accounts receivable (AR) and overdue accounts receivable (OPR) for one client as of the beginning of the specified week:

Rice. 1. Initial data

The share of PD in the total PD was selected as the parameter that is planned to be monitored. Since the level of business fluctuates throughout the year, it is more logical to use a relative parameter, since absolute numbers will reflect not only the client’s payment discipline, but also the level of business.

Data by week, as well as the control limit, are plotted on the control chart. The latter is equal to µ + 3σ, where µ is the average value and σ is the standard deviation. You can use µ and σ determined from the first 10–15 values. I prefer to use sliding values of µ and σ, determined over all values. Such µ and σ will change when new values are added corresponding to new weeks.

To control accounts receivable, the lower control limit is not used, since the lower the value, the better. If you exercise control over some technical parameter, then in this case the lower limit also has a physical meaning and should be plotted on the graph. For clarity, I also like to plot the mean line on the control charts (Figure 2). In principle, this is not necessary...

Rice. 2. Shewhart's checklist for accounts receivable management.

Why do the control limits correspond to the values of µ ± 3σ? In accordance withShewhart's conceptIt is precisely this definition of boundaries that makes it possible to separate situations when economically feasible begin searching for special causes of variation; As long as such limits are not exceeded, the process remains statistically controllable, and the search for reasons for the deviation of individual values is economically unfeasible. That is, one should not look for an answer [to the question of why µ ± 3σ] in probability theory or statistical analysis.

Let me emphasize once again: defining the values of µ ± 3σ as boundaries reflects only the practical usefulness of just such a definition. An important conclusion follows from this: in each specific case it makes sense to pay attention to deviations beyond the limits of µ ± 2σ, which can also be due to special causes of variations (simply probability the fact that such deviations are associated with special causes of variations is lower than in the case of going beyond µ ± 3σ). Should managers take any measures if they go beyond µ ± 2σ!? The question is subtle. Personally, I limit myself to informing those responsible that the situation is close to problematic, and ask them to discuss it with the client...

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Introduction

The traditional approach to manufacturing, regardless of product type, is manufacturing and quality control to inspect finished products and reject units that do not meet specifications. This strategy often leads to losses and is not economical, since it is based on post-factum testing, when defective products have already been created. A more effective loss prevention strategy is to avoid producing unusable products. This strategy involves collecting information about the processes themselves, analyzing it and taking effective actions in relation to them, and not to the products.

A control chart is a graphical tool that uses statistical approaches whose importance in process control was first demonstrated by Dr. W. Shewhart in 1924.

The purpose of control charts is to detect unnatural variations in data from repeated processes and to provide criteria for detecting a lack of statistical control. The process is in a statistically controlled state if the variability is caused only by random reasons. In determining this acceptable level of variability, any deviation from it is considered to be the result of special causes that must be identified, eliminated or mitigated.

The task of statistical process control is to ensure and maintain processes at an acceptable and stable level, ensuring that products and services meet established requirements. The main statistical tool used for this purpose is the control chart, a graphical way of presenting and comparing information based on a sequence of samples reflecting the current state of the process with boundaries established on the basis of the inherent variability of the process. The control chart method helps determine whether a process has actually reached or remains in a statistically controlled state at a properly specified level, and then maintain control and a high degree of uniformity of the critical characteristics of a product or service by continuously recording product quality information during the production process.

The use of control charts and their careful analysis lead to better understanding and improvement of processes.

1. Statistical methods for product quality management

1.1 The role of statistical control methods

The main objective of statistical control methods is to ensure the production of usable products and the provision of useful services at the lowest cost. For this purpose, analyzes of new operations or other studies aimed at ensuring the production of usable products are carried out.

The introduction of statistical control methods gives results in the following indicators:

1. improving the quality of purchased raw materials;

2. saving of raw materials and labor;

3. improving the quality of manufactured products;

4. reduction in the number of defects;

5. reduction of control costs;

6. improving the relationship between production and consumer;

7. facilitating the transition of production from one type of product to another.

One of the basic principles of quality control using statistical methods is the desire to improve product quality by monitoring various stages of the production process.

Depending on the goals set for product quality management at enterprises, statistical methods can be used for:

Statistical analysis of the accuracy and stability of products, technological processes, equipment, etc.;

Statistical regulation and management of technological processes;

Statistical acceptance control of product quality and its evaluation.

Statistical analysis of the accuracy and stability of technological processes - establishing by statistical methods the values of indicators of accuracy and stability of the technological process and determining the patterns of its occurrence over time.

Determine the actual value of indicators of accuracy and stability of the technological process, equipment or product quality;

Identify the degree of influence of random and systematic factors on the accuracy and stability of the technological process and product quality;

Justify technical standards and approvals for products;

Identify reserves of the production and technological process;

Justify the choice of technological equipment and measuring instruments for manufacturing products;

Identify the possibility and justify the feasibility of introducing statistical methods into the production process;

Assess the reliability of technological systems;

Justify the need for reconstruction of the technological process or repair of technological equipment and other measures to improve the technical process;

During periodic checks of the technological accuracy of equipment and accessories in the process of monitoring compliance with the technological discipline of manufacturing products of the main production;

When carrying out in-plant certification of technological processes;

When installing new technological equipment and accepting equipment after repair;

When analyzing and evaluating indicators of the production process and product quality, etc.

In the conditions of serial, small-scale and pilot production, statistical analysis is primarily recommended to be implemented for the systematic assessment of the accuracy of technological equipment and the rational placement of work on this equipment.

1.2 Shewhart control charts

The control chart is a special form on which a central line and two lines are drawn: above and below the average, called the upper and lower control limits. The data of measurements and control of parameters and production conditions are plotted on the map with dots.

When examining changes in data over time, you should ensure that the graph points do not go beyond the control limits. If an outlier of one or more points beyond the control limits is detected, this is perceived as a deviation of the parameters or process conditions from the established norm.

To identify the cause of deviation, the influence of the quality of the source material or parts, methods, operations, conditions for carrying out technological operations, and equipment is examined.

In production practice, the following types of control charts are used:

1. map of arithmetic averages and ranges: -R is used in the case of control on a quantitative basis, such quality indicators as length, weight, tensile strength, etc.

2. map of arithmetic averages and standard deviations: -S map is similar to the -R map, but has a more accurate map of process variability and is more complex to construct.

3. map of medians and ranges: -R map is used for the same situations as -R maps, the advantage is the absence of complex calculations, but the median map is less sensitive to changes in the process.

4. map of individual values: X-map is used when it is necessary to quickly detect undetected factors or in cases where only one observation was made in one day or week.

5. map of the share of defective products: p-map - used in the case of control to determine the share of defective products.

6. map of the number of defective units of production: np-map - used in the case of control to determine the number of defective products.

7. map of the number of defects: the c-card is used in the case when quality control is carried out by determining the total number of defects in a predetermined constant volume of inspected products.

8. map of the number of defects per unit of product: u-map - used in the case of quality control by the number of defects per unit of product, when the area, length or other parameter of the product sample is not a constant value.

The data presented in the control chart is used to construct histograms; the graphs obtained on the control charts are compared with control standards. All this allows you to obtain valuable information to solve problems that arise.

2. Initial data, goals and objectives

The purpose of the work is to analyze the technological process using Shewhart control charts and prescribe appropriate measures and recommendations if an uncontrollable state of the process is detected.

To achieve this goal, certain tasks should be solved step by step, which include:

Selecting the type of control charts, taking into account the specifics of their application;

Processing a data array, carrying out the necessary calculations and constructing control charts;

3. Construction and analysis of control charts

3.1 Selecting the type of control charts

Shewhart's control charts are divided into quantitative and qualitative (alternative) depending on the measurability of the indicator being studied. If the value of the indicator is measurable (temperature, weight, size, etc.), maps of the indicator value, ranges and double Shewhart maps are used. On the contrary, if the indicator does not allow the use of numerical measurements, use map types for an alternative indicator. In fact, the indicators studied on this basis are determined as meeting or not meeting the requirements. Hence the use of maps for the proportion (number) of defects and the number of conformities (non-conformities) per unit of production.

To determine the most suitable control chart for the data array under consideration, we will use the algorithm presented in Figure 3.1.

Figure 3.1 - Algorithm for selecting control cards

Based on the algorithm presented above, it follows that at the first stage we should determine what type of data about the process we receive.

There are two types of control charts: one is designed to control quality parameters, which are continuous random variables, the values of which are quantitative data of the quality parameter (dimensional values, weight, electrical and mechanical parameters, etc.). And the second is for monitoring quality parameters, which are discrete (alternative) random variables and values that are qualitative data (pass - fail, conform - do not conform, defective - defect-free product, etc.).

In this work, an array of quantitative data on the quality parameter is considered; based on this, at the next stage, the choice of a control chart depends on the sample size, their number and the conditions for constructing the control chart.

Maps for quantitative data reflect the state of the process through dispersion (variability from unit to unit) and through the location of the center (process average). Therefore, control charts for quantitative data are almost always used and analyzed in pairs—one chart for location and one for scatter. The most commonly used pair is the - and R - card.

Card type - R is used in mass production, when type X cards are not applicable due to bulkiness. When using type - R cards, conclusions about the stability (sustainability) of the process are made on the basis of data obtained from the analysis of a small number of representatives of all the products under consideration. In this case, all products are combined into batches in the order of manufacture and small samples, no more than 9, are taken from each batch, based on the data of which a control chart is constructed.

Control chart of individual values (X) - this chart is used if observations are made on a small number of objects, and all of them are subject to control. Observations are made on a continuous indicator.

When using individual value maps, rational subgrouping is not used to provide an estimate of within-batch variability and control limits are calculated based on a measure of variation obtained from sliding ranges, usually of two observations. The sliding range is the absolute value of the difference in measurements in successive pairs, i.e. the difference between the first and second dimensions, then the second and third, etc. Based on the moving ranges, the average moving range is calculated, which is used to construct control charts. The overall average is also calculated for all data.

Median maps are an alternative to R maps for process control with measured data. They provide similar findings and have certain advantages. Such cards are easy to use and do not require large calculations. This may make it easier to introduce them into production. Because median values are plotted along with individual values, a median map provides a scattering of process results and a detailed picture of variation.

Control chart of means and standard deviations (-S). This map is almost identical to the (-R) map, but is more accurate and can be recommended for debugging technological processes in the mass production of critical parts. It can be used in cases where there is a built-in control system with automatic data entry into a computer used for automatic process control.

In maps - S, instead of the range R, a more effective statistical characteristic of the dispersion of observed values is used - the standard deviation (S). It shows how closely individual values cluster around the arithmetic mean or how they scatter around it.

Analyzing the initial data array, we note that the number of samples is 15, the volume of each is 20. Also, when choosing a control chart, we will take into account the need for speed in constructing control charts and ease of calculations. Based on this, we will conclude about the most suitable type of control charts for a quantitative trait.

Since we have a sample size of more than 9, we have the necessary resources to carry out complex calculations (Microsoft Excel is used in this work), we will use the most accurate type of control charts for a quantitative characteristic, namely S charts.

3.2 Calculation and construction of control charts

The procedure for constructing an S map, conditionally, can be divided into several stages:

Calculation of the mean (and standard deviation of each sample (S);

Calculation of average lines for - map (), and S - map;

Calculation of control limits for the map (UCLX and LCLX), for map S (UCLS and LCLS);

Drawing of the center line, average sample values, control limits and technological tolerance limits on the map.

Drawing on the S-map the average line, standard deviations of each sample and control limits.

The sample mean (and standard deviation S is calculated using the formulas:

where: X - parameter value; n - sample size.

Substituting the sample values into formulas 3.1 and 3.2, we calculate the average value and standard deviation for each sample (Table 3.1).

Table 3.1 - Results of calculating average values and square deviations of samples

Sample no.

To calculate the average lines and S maps, we will use formulas 3.3 and 3.4.

where, k is the number of subgroups.

Substituting the data from table 3.1 into formulas 3.3 and 3.4, we get:

The obtained values of the middle lines are necessary for calculating control limits, which are calculated using the formulas:

UCLX = + A3 H; (3.5)

LCLX = - A3 H; (3.6)

UCLS= V4 H; (3.7)

LCLS= V3 H; (3.8)

where: A3, B4, B3 - coefficients for calculating control limits.

Coefficients for calculating control limits are presented in GOST R 50779.42-99 “Statistical methods. Shewhart's control charts." Based on this standard, we select the coefficients necessary for calculations:

Let's calculate the numerical values of the control limits by substituting the necessary values:

UCLX = 8.943833+0.68Х0.912466=9.56431;

LCLX = 8.943833 - 0.68Х0.912466= 8.323356;

UCLS= 1.49Х0.912466= 1.359575;

LCLS= 0.51Х0.912466= 0.465358;

All calculations and transformations of the original data array were carried out in Microsoft Excel.

An array of control results values together with calculation results is registered in a special form.

When constructing control charts, you need to pay attention to the choice of scales. For each type of control chart, the difference between the upper and lower value of the scale and the value of the scale division will be different.

In the case of constructing an S map, the following features should be noted when choosing scales:

For a map, the difference between the top and bottom values of the scale should be approximately twice the difference between the highest and lowest values of the subgroup means;

For the S map, the scale should have values from 0 to twice the maximum value of S in the initial period (5-6 first subgroups);

Scales and S cards must have the same value of divisions.

Thus, guided by the above, we will determine the maximum and minimum values of the scales for control charts.

The maximum and minimum values of the subgroup means are 9.62 and 8.64, respectively, the doubled difference between these values is ~1.25. Since the difference between the largest and smallest technological tolerance values is much greater, we are forced to expand the range of scale values to 7.40 and 11.20, respectively.

The maximum value of the standard deviation in the initial period is 0.98, doubling this number, we get the maximum value of the scale - 1.96. Thus, for card S the range of scale values is from 0 to 2. The scale division price for and S cards will be equal to 0.2. The construction of control charts was also carried out using Microsoft Excel tools.

3.3 Control chart analysis

The goal of this step is to recognize indications that the variability or mean is not remaining at a constant level, that one or both are out of control, and that appropriate action is needed.

The purpose of a process control system is to obtain a statistical signal about the presence of special (non-random) causes of variation. Systematic elimination of special causes of excess variability brings the process into a state of statistical control. If the process is in a statistically controlled state, the quality of the product is predictable and the process is suitable to meet the requirements established in the regulatory documents.

The Shewhart chart system is based on the following condition: if the variability of the process from unit to unit and the process average remain constant at given levels (estimated by S and X), then the deviations S and the average X of individual groups will change only randomly and rarely go beyond the control limits . Obvious trends or patterns in the data are not allowed, other than those that occur by chance with some degree of probability.

Exit from the controlled state is determined by the control chart based on the following criteria:

1) Points going beyond control limits.

2) A series is a manifestation of a state where the points invariably end up on one side of the midline; the number of such points is called the length of the series.

A series of seven dots is considered non-random.

Even if the length of the series turns out to be less than six, in some cases the situation should be considered non-random, for example, when:

a) at least 10 out of 11 points are on one side of the center line;

b) at least 12 out of 14 points are on one side of the center line;

c) at least 16 out of 20 points are on one side of the center line.

3) Trend (drift). If the points form a continuously rising or falling curve, it is said to be trending.

4) Approaching control “zones” limits. Points that approach the 3-sigma control limits are considered, and if 2 or 3 points are outside the 2-sigma lines, then such a case should be considered abnormal.

5) Approaching the center line. When the majority of points are concentrated within the middle third, due to an inappropriate method of dividing into subgroups. Approaching the central line does not mean that a controlled state has been achieved; on the contrary, it means that data from different distributions are mixed in subgroups, which makes the range of control limits too wide. In this case, you need to change the method of dividing into subgroups.

The S and maps are analyzed separately, but a comparison of the course of their curves can provide additional information about the special reasons for the impact on the process.

On a standard deviation map, a point above UCLS could mean:

Part-to-part variability has increased, either at one point or as part of a trend;

The measuring system has lost proper resolution.

A point below LCLS on the standard deviation map could mean:

Incorrect calculation of the control boundary or incorrect marking of the point;

Part-to-part variability has decreased;

The measuring system has changed;

A series of dots above or an increasing series of dots can mean:

The scatter of the value has increased, which could have occurred due to an irregular reason;

Changes in the measuring system;

A series of dots below or a decreasing series of dots can mean:

The spread of values has decreased, which is a positive factor that must be used to improve the process;

There has been a change in the measuring system.

Non-random behavior of points, manifested in the form of shifts, trends, and cyclicity, is also possible.

To analyze control charts for points approaching the midline, it is necessary to calculate the boundaries of the middle third.

To calculate the middle third, we introduce coefficient A, which is equal to a third of the difference between the value of the upper control border of the map and the value of its middle line (formula 3.9).

A=(UCL-CL)/3; (3.9)

Where: UCL - upper control limit; CL - midline value; A is the coefficient.

The boundaries of the middle third are calculated using the formulas:

VGST=CL+A; (3.10)

NGST=CL-A; (3.11)

Where: VGST - the upper limit of the middle third; NGST - lower limit of the middle third; Let's calculate coefficient A for cards and S:

Ах= (9.56-8.94)/3= 0.207;

АS= (1.36 - 0.91)/3= 0.149.

Substituting the values into formulas 3.10 and 3.11, we obtain the values of the upper and lower boundaries of the middle third, respectively:

VGSTx=8.94+0.207= 9.15;

VGSTS=0.91+0.149= 1.06;

NGSTx=8.94-0.207= 8.74;

NGSTS=0.91-0.149= 0.76;

The boundaries of the middle third are also included in the calculation results table.

Analyzing the obtained control charts, we will draw up a table in which we will describe the state of controllability of the process based on the above criteria.

Table 3.2 - Analysis of control charts

Criterion
Points above UCL			The absence of points beyond the control limits indicates the stability of the process. Its variability is also stable, which is a positive factor.
Points below LCL			The absence of points beyond the control limits indicates the stability of the process.
			On the map, starting from points 11 to 15, a shift in the process is observed. A shift in the points may mean that the points have begun to cluster around a new mean value.
			There is no cyclicity in the location of the points. The absence of such behavior of the points indicates that there are no reasons that can periodically influence the process (work shifts, time of day).
			On map S there is a slightly increasing trend starting from point 9. This means that the spread of values is gradually increasing, which is not a positive factor.
Series of points			Note points 6 through 11 on the average map. There is a series of dots above the midline.
Scatter of points within the middle third			This percentage of points falling into the middle third is considered normal.

After identifying non-standard behavior of points on maps, it is necessary to find the reason for their appearance and introduce corrective actions.

A slightly increasing trend on the S map may be caused by changes in the measurement system, incompetence of personnel, or equipment malfunction. Due to the small number of points, it is necessary to continue observations. If non-standard behavior of points is confirmed, it is necessary to identify the cause and introduce corrective actions.

To identify the reasons, take the following steps:

Technical inspection of equipment;

Calibration, verification of measuring instruments;

Checking the qualifications of the worker performing the operation;

Checking the controller's competence.

Corrective actions may include:

Shifting points on the average map can be caused by changes in the measurement system, wear, or equipment failure. Due to the small number of points, the analysis should be continued to identify the reasons for this arrangement of points. If the assumptions about the occurrence of a shift are confirmed, it is necessary to identify the cause and prescribe appropriate corrective actions.

A series of points on the map may indicate changes in the process associated with equipment, measuring systems, and workers. There is a series of points 6 to 11 on the average map. The measuring system should be checked for changes in a given period of time, the competence of the worker performing the operation, the equipment and the appropriate corrective actions should be introduced:

Adjustment, configuration, repair or replacement of equipment;

Improvement of personnel qualifications, improvement of working conditions;

Adjustment, adjustment, repair or replacement of measuring instruments.

Process maps allow you to monitor the process and identify non-standard changes in process parameters while still within technological tolerances.

Analysis of process maps helps to identify non-random causes affecting the process. Such causes must be eliminated; systematic elimination of special causes of excessive variability brings the process into a state of statistical controllability. If the process is in a statistically controlled state, the quality of the product is predictable and the process is suitable to meet the requirements established in the regulatory documents.

After bringing the process into a statistically controlled state, it becomes possible to evaluate the technological capabilities of the process. The process is first brought into a statistically controlled state, and then its capabilities are determined. Thus, the determination of process capabilities begins after the control tasks using - and S - cards are solved, i.e. special causes are identified, analyzed, corrected and their recurrence is prevented. Current control charts must demonstrate that the process remains in statistical control for at least 25 subgroups.

As a guide to action, you can use the procedure schematically presented in Figure 3.2.

Figure 3.2 - Process improvement strategy

Conclusion

statistical production mean square shewhart

The quality of products (works, services) is decisive in the public assessment of the performance of each work team. The release of effective and high-quality products allows the enterprise to receive additional profit and ensure self-financing of production and social development.

Shewhart control charts as a tool for quality control of processes and products are successfully used at many enterprises, including Russian ones.

Control charts have become widespread due to their ability to prevent defects. This state of affairs helps to significantly reduce production costs associated with the production of non-conforming products.

This paper provides an example of the use of Shewhart control charts for process control. In the course of the work, the original data array was transformed, control charts were selected taking into account their features. As a result of the selection, the most preferred card for this task is the -S card.

The work on carrying out the necessary calculations and construction was carried out using Microsoft Excel.

As a result of the analysis of control charts, the following non-standard situations for the location of points were identified:

Slightly increasing trend on map S;

Possible process shift on the map;

A series of dots above the center line on a map.

The actions necessary to bring the process into a statistically controlled state were assigned.

List of sources used

1. GOST R 50779.0-95 Statistical methods. Basic provisions.

2. GOST R 50779.11-2000 Statistical methods of quality management. Terms and Definitions.

3. GOST R 50779.42-99 Statistical methods. Shewhart control charts.

4. Efimov V.V. Quality management tools and methods: textbook / V.V. Efimov. - 2nd ed., erased. - M.: KNORUS, 2010. - 232 p.

5. Tsarev Yu.V., Trostin A.N. Statistical methods of quality management. Control cards: Educational and methodological manual / State Educational Institution of Higher Professional Education Ivan. state chem. - technol. univ. - Ivanovo, 2006.- 250 p.

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ST. PETERSBURG STATE UNIVERSITY

FACULTY OF ECONOMICS

Department of Economics and Enterprise Management

Shewhart control charts in a quality management system

Course work

2nd year students of the EUP group - 22

day department

specialty 080502 - “Economics and enterprise management”

Scientific adviser:

Saint Petersburg

Introduction

Chapter 1. Concept of a quality management system

Chapter 2. The importance of statistical methods in quality management

Chapter 2.1. Shewhart control charts as a method of statistical control and quality management

Chapter 3. Construction of Shewhart control charts

Conclusion

Literature

Annex 1

The peak of development of quality management occurred in 1980-1990, when the quality management system was widely introduced. In its early days, the concept helped many companies rethink their product manufacturing process and avoid multimillion-dollar costs associated with producing defective products.

In parallel with reducing the number of defects and improving product quality, companies began to pay more attention to consumers and their desires. After all, as you know, attracting a new client can cost a company 6 times more than retaining an existing one.

In the early stages of its development, quality management was not very different from careful administration or dispatching, but as time passed, the theory developed and the practice of applying the concept expanded. Now, not only industrial, but also service companies practice a quality approach and use modern quality control tools; As a rule, these are automated systems (ERP, MRP, process control systems) that have in their arsenal applications for creating diagrams, maps, recording the number of defects, or simply conveniently organizing customer data (CRM).

The purpose of this work is to systematize knowledge in the field of quality management. This determined the structure of the course work; the first chapter is devoted to considering the historical aspects of the development of the concept; a description of the significance of statistical methods - the second chapter; and the construction of control charts, using the example of a random sample of a certain process - in the third. The consideration of Shewhart's control charts, and not other, later developments, is explained, first of all, by the fact that Shewhart's work gave impetus to the development of the concept in this direction. And for a deeper understanding of the entire quality management, it is necessary to have knowledge about the emergence of significant discoveries.

Quality management has many definitions, depending on the position taken by the author. Some highlight the special role of the human factor, others - the importance of a systematic approach and quantitative measurements, while others emphasize the evolution of management schools.

So, quality management, in a broad sense, is the management of an enterprise that allows you to most fully satisfy the needs of customers and anticipate their expectations. Naturally, in my opinion, questions arise: firstly, how is their satisfaction achieved, and secondly, how does the quality management approach in this regard differ from the usual process of product planning and production?

Answering the question about consumer satisfaction, we can say that quality management takes the consumer’s attitude to the quality of the resulting product as the main condition. In this case, product quality becomes the most significant indicator for the consumer and, as a result, the main competitive advantage.

The second question concerns the differences between conventional production and one where quality principles are applied. An interesting position is taken by Japanese authors who attribute the process of product quality management to a special philosophy of the enterprise, a new view of production and inextricably linked with the concept of continuous improvement. In addition to this slightly idealized attitude, another difference can be shown; The normal production process involves a number of activities aimed at identifying and satisfying customer needs, which is also stated in the definition of quality management. However, the qualitative approach emphasizes the inherent importance of producing quality products, at all stages of production, from product development to timely delivery to the consumer. This approach dictates the priority task facing the enterprise - the production of high-quality products from cycle to cycle, which undoubtedly guarantees the consistency of the consumer receiving good products. For an enterprise, this, first of all, means gaining the respect of consumers and developing their loyalty, which in modern conditions is far from an unimportant characteristic.

To summarize, we see that consumers receive high-quality products, and manufacturers receive stable profits. Modern markets show a rapid pace of development, which sets a condition for firms: “develop to survive.” And in this case, good, high-quality products, but not meeting market requirements, will also not be able to provide significant competition, like a company whose 30% of products are defective goods. That is why quality management plays an important role in anticipating the expectations and needs of the consumer, creating new needs for him and satisfying them, in accordance with the approach to ensuring product quality.

As shown above, quality management is an extensive process, affecting the entire production, all levels of management (from controllers to senior managers) and all production processes. But where and under what conditions did it originate? What contributed to the emergence of a new approach to management? Let's look at quality management in retrospect.

Product quality management runs through the entire history of management development as a red line. Starting from Towne’s famous 1866 work “The Engineer as Economist,” it is customary to talk about the origins of management.

Inspired by Towne's work, F. Taylor became the founder of the scientific school of management. His approach literally revolutionized manufacturing. In addition to introducing the practice of measuring the time spent on various operations, Taylor established requirements for the quality of products in the form of tolerance fields (pass and fail gauges). He also established a system of fines for defects (up to and including dismissal), motivation and training of employees. Taylor's revolutionary approach gave impetus to the further development of management.

Another not unknown manager of the 20th century was Henry Ford, who founded the car company that still exists today. By developing the Model T, Ford doomed himself to perpetuation. He not only invented a lightweight, durable (for those times) and unpretentious car, but also introduced a system of mass assembly line production. He unified and standardized all operations, and included after-sales service in the production area. He became involved in labor protection and the creation of normal working conditions. “According to Henry Ford, the main factor in the success of an enterprise is the quality product it produces. Until the quality is proven, the production of the product cannot begin.”

Emerson made a major contribution to the development of management with his book, The 12 Principles of Productivity, published in 1912. Emerson noted the importance of goal setting, scheduling, rewards for performance, and other principles. He saw efficiency as a key aspect of production organization, by increasing which it is possible to achieve high results while avoiding overexertion.

With the further development of management, enterprises faced the need to reduce labor costs for quality control, since previous methods of quality control, which involved monitoring each unit of output, led to an increase in the number of inspectors. The problem was solved by the methods that replaced them - methods of statistical quality control. G. Dodge and G. Roming proposed methods of sampling, which made it possible to check not all products, but a certain amount from the entire batch. Statistical control was carried out by new specialists - quality engineers.

A major contribution to the application of statistical methods belongs to Walter Shewhart, who, while working at Bell Telephone Laboratories (now AT&T) as part of a group of quality specialists, in the mid-1920s. laid the foundations of statistical quality control. Shewhart is considered one of the patriarchs of modern philosophy of quality. Shewhart paid much attention to the compilation and analysis of control charts, which will be discussed in subsequent chapters.

The great contribution of Edward Deming, an American expert in the field of quality. During World War II, he trained US engineers in quality control as part of the national defense program. After the war, in 1950, Deming was invited to occupied Japan to present a theory jointly with Shewhart. Speaking to the owners and managers of most enterprises, Deming exhorted that if statistical methods were followed, then very soon Japanese manufacturers would be able to enter world markets. Which was vital for post-war Japan.

Deming's teachings set the direction for the development of Japanese companies. Deming inspired the public with his ideas, “no nation needs to be poor” was his opening phrase. Very soon, Japan entered world markets with goods superior in quality to their American and European counterparts.

The next scientist to come to Japan from America was Juran. Juran considered quality issues at the level of the entire company and individual divisions. Juran's lectures were practical in nature, and the emphasis was placed on determining indicators of quality products, establishing standards and methods of measurement, and compliance of products with specifications.

The goal of the quality approach is to create a better product that can better meet customer needs. And such a complex problem cannot be solved only by carrying out the necessary measurements and analyzing the data obtained. To achieve such a goal, it is sometimes necessary to modernize existing equipment, improve the production process, or change it entirely. It is also worth considering the necessary work before (marketing research, design, procurement) and after (packaging, storage, delivery, sales and after-sales service) the production of products. All this proves the need to consider quality management in a single system and manage it, adhering to one strategy throughout the enterprise.

In parallel with Deming and Juran, Dr. Feigenbaum (USA), in the 50s, in the monograph “total quality management”, sets out the importance of a systemic (integrated) approach to product quality management.

In 1922, an expert group from the USA coined the concept of Total Quality: “Total quality (TQ) is a management system focused on people, the goal of which is to constantly increase the degree of customer satisfaction while constantly reducing real costs. TQ is a system-wide approach (rather than individual areas or programs) and an integral part of top-level strategy; it works horizontally across functions and departments, involving all employees from the top down, and going beyond traditional boundaries to include both the supply chain and the consumer chain. In TQ, great emphasis is placed on mastering the policy of constant change and its adaptation, since these components are considered powerful levers that significantly influence the success of the organization."

The next stage in the development of a quality management system is the development of a process approach and the popularization of reengineering. Reengineering proposes to replace the principle of division of labor in management with a process approach. At the head of the organization are processes that have their own executors. Enterprises were embraced by a new idea, a massive revision of the operation of processes began, their optimization, changes and the introduction of new ones. Until it was discovered that reengineering is by no means a universal remedy.

Now, in the 21st century, the adaptive model of organization is taking root in science and the concept of knowledge management is spreading.

But despite the widespread knowledge of quality management methods and systems, many enterprises do not realize the importance of quality control. In an effort to keep up with world standards, they install software products and build control charts, without understanding how this can help them.

No matter how simple or complex the quality management methods are, they alone will not be able to provide any benefit to the enterprise, because even after conducting all the necessary research and obtaining conclusions, changes must still be developed and implemented. A significant part of Russian enterprises, when starting to develop a quality management system (QMS), do not set the goal of achieving effectiveness, and especially the effectiveness of the QMS, which is a prerequisite for quality management. The implementation of the widespread ISO system is more reminiscent of expensive certification than management aimed at customer satisfaction.

The introduction of total quality management in Russia is associated with significant difficulties, and first of all, this is the rejection of the concept of quality by managers, the unwillingness to be leaders committed to the implementation of quality and to follow the chosen goal.

The specifics of Russia, its people, morals and orders, apparently, will not soon be ready for fundamental changes in the system of views on the management of an organization.

These are the main milestones in the development of product quality management systems.

Shewhart card quality management

The importance of statistical methods can hardly be overestimated, since without such control methods it would be difficult, almost impossible, to identify the dependence of defects on certain factors. At the same time, organizations should strive to reduce the variability of factors, and as a result, demonstrate greater stability of product quality. For example, during the machining of metal, a cutter is used, which becomes slightly dull after processing a new piece of metal. In addition, changes in temperature, composition of the cutting fluid, or the influence of other factors can lead to defective products.

Not all factors involved in production are constant; statistical methods of quality control and management are aimed at reducing their variability. There are, however, other ways to reduce product defect rates, such as using expert intuition or past experience in resolving similar problems.

Let's consider statistical methods of quality control. Kaeru Ishikawa, emeritus professor at the University of Tokyo, proposed dividing statistical methods into three groups:

1. elementary methods, these include the “seven simple tools of quality”

check sheet

æ allows you to record data on defects encountered by the controller in a convenient form. In the future, it becomes a source of statistical information.

quality histogram

æ It is built on the basis of a control sheet and shows the frequency of the values of the controlled parameter falling within specified intervals.

cause-and-effect diagram

æ is also called a fishbone diagram. The diagram is based on one quality indicator, which takes the form of a straight horizontal line (“ridge”), to which the main reasons influencing the indicator (“big bones of the ridge”) are attached by lines. Secondary and tertiary causes that influence older causes are also connected by straight lines (“medium and small bones”). After construction, it is necessary to rank all the reasons according to the degree of influence on the indicator.

Pareto chart

æ The main assumption of the diagram is that in most cases, the overwhelming number of defects arises due to a small number of important reasons.

The consequence of the sharpened diagram will be the conclusion about which types of defects have a larger share among the others and, accordingly, what you should pay special attention to.

·Stratification

æ Stratification or stratification of data is carried out when it is necessary to compare the results of similar processes performed by different workers, or on different machines, using different materials, and in other cases.

Scatter diagram

æ is built on the basis of paired data (for example, the number of defects on the air temperature in the furnace), the dependence of which must be studied. The diagram can provide information about the shape of the distribution of pairs. Based on the diagram, it is possible to conduct correlation and regression analysis.

control card

æ the principles and methods of constructing control charts will be discussed in the third chapter of the work.

2. intermediate methods, these are acceptance control methods, distribution theories, statistical estimates and criteria.

3. advanced methods are methods based on the use of computer technology:

·experiment planning,

multivariate analysis

Product quality is determined by a set of values and characteristics, which in general can be called quality indicators. Statistical studies are carried out on their basis. Indicators characterize the consumer properties of products and can have different meaningful meanings.

Control charts belong to the “seven simple methods” of quality management, according to K. Ishikawa’s classification. Like other methods, control charts are aimed at identifying factors that influence process variability. Since variability can be influenced by random or certain (non-random) reasons. Random reasons include those whose occurrence cannot be avoided, even using the same raw materials, equipment and workers servicing the process (an example would be fluctuations in ambient temperature, material characteristics, etc.). Certain (non-random) reasons imply the presence of some relationship between changes in factors and process variability. Such causes can be identified and eliminated when setting up the process (for example, loose fasteners, tool wear, insufficient sharpening of the machine, etc.). In an ideal situation, the variability of certain factors should be reduced to zero, and by improving the technological process, the influence of random factors should be reduced.

Control charts are used to adjust existing processes to ensure that products meet specifications.

The construction of control charts is mainly aimed at confirming or rejecting the hypothesis about the stability and controllability of the process. Due to the fact that the maps are multiple in nature, they make it possible to determine whether the process under study is occurring randomly; if so, then the process should tend to a normal, Gaussian distribution. Otherwise, trends, series and other abnormal deviations can be traced on the graph.

The next chapter will cover the practical part regarding Shewhart control charts.

Before proceeding with the actual construction of control charts, let’s get acquainted with the main stages of the task. So, due to the fact that different authors pursue their own goals when describing the construction of control charts, below will be presented an original vision of the stages of constructing Shewhart’s control charts.

Algorithm for constructing Shewhart control charts:

I. Process analysis.

First of all, it is necessary to ask about the existing problem, because, in the absence of these, the analysis will not make sense. For greater clarity, you can use the Ishikawa cause-and-effect diagram (mentioned above, Chapter 2). To compile it, it is recommended to involve employees from different departments and use brainstorming. Having carried out a thorough analysis of the problem and found out the factors influencing it, we move on to the second stage.

II. Process selection.

Having clarified the factors influencing the process in the previous stage and drawn a detailed skeleton of the “fish”, it is necessary to select a process that will be subject to further research. This step is very important because choosing the wrong indicators will make the entire control chart less effective due to the study of insignificant indicators. At this stage, it is worth recognizing that the choice of the appropriate process and indicator determines the outcome of the entire study and the costs associated with it.

Here are some examples of possible indicators:

Table 1. Application of control cards in service organizations

Source Evans J. Quality management: textbook. Allowance/J. Evans.-M.: Unity-Dana, 2007.

At the same time, the indicator should be chosen based on the main goal of the company, namely, meeting the needs of customers. When a process and an indicator characterizing it have been selected, you can proceed to data collection.

III. Data collection.

The purpose of this stage is to collect data about the process. To do this, it is necessary to design the most suitable method for collecting data, find out who will take measurements and at what time. If the process is not equipped with technical means to automate the entry and processing of data, it is possible to use one of Ishikawa’s seven simple methods - checklists. Control sheets, in fact, are forms for recording the parameter being studied. Their advantage lies in ease of use and ease of employee training. If there is a computer at the workplace, it is possible to enter data through the appropriate software products.

Depending on the specifics of the indicator, the frequency, time of collection and sample size are determined to ensure the representativeness of the data. The collected data is the basis for further operations and calculations.

After collecting information, the researcher must decide whether to group the data. Grouping often determines the performance of control charts. Here, with the help of the analysis already carried out using a cause-and-effect diagram, it is possible to establish the factors by which the data can be most rationally grouped. It should be noted that data within one group should have little variability, otherwise the data may be misinterpreted. Also, if the process is divided into parts using stratification, each part should be analyzed separately (example: production of identical parts by different workers).

Changing the method of grouping will lead to a change in the factors that form within-group variations. Therefore, it is necessary to study the factors influencing the change in the indicator in order to be able to apply the correct grouping.

IV. Calculation of control chart values.

For any type of Shewhart chart, it is assumed that the central and control lines are determined, where the central line (CL-controllimit) actually represents the average value of the indicator, and the control limits (UCL-uppercontrollimit; LCL-lowercontrollimit) are the permissible tolerance values.

The values of the upper and lower control limits are determined by formulas for different types of maps, as can be seen from the diagram in Appendix 1. To calculate them, in order to replace cumbersome formulas, coefficients from special tables for constructing control maps are used, where the value of the coefficient depends on the sample size (Appendix 2). If the sample size is large, then maps are used that provide the most complete information.

At this stage, the researcher must calculate the values of CL, UCL, LCL.

V. Construction of a control chart.

So, we come to the most interesting process - a graphical reflection of the data obtained. So, if the data was entered into a computer, then using the Statistica or Excel program environment, you can quickly graphically display the data. However, it is possible to construct a control chart and, without having special programs, then, along the OY axis of the control charts we plot the values of the quality indicator, and along the OX - the moments in time of recording the values, in the following sequence:

1. draw the center line (CL) on the control card

2. draw boundaries (UCL; LCL)

3. reflect the data obtained during the study by placing the appropriate marker at the intersection point of the indicator value and the time of its registration. It is recommended to use different types of markers for values within and outside the tolerance limits.

4. in case of using double cards, repeat steps 1-3 for the second card.

VI. Checking the stability and controllability of the process.

This stage is designed to show us what the research was carried out for – whether the process is stable. Stability (statistical controllability) is understood as a state in which repeatability of parameters is guaranteed. Thus, the process will be stable only if the following cases do not occur.

Let's consider the main criteria for process instability:

1. Exceeding control limits

2. Series – a certain number of points that invariably appear on one side of the central line - (top) bottom.

A series of seven dots is considered abnormal. In addition, the situation should be considered abnormal if:

a) at least 10 out of 11 points are on one side of the center line;

b) at least 12 out of 14 points are on one side of the center line;

c) at least 16 out of 20 points are on one side of the center line.

3. trend – a continuously rising or falling curve.

4. approaching control limits. If 2 or 3 points are very close to the control limits, this indicates an abnormal distribution.

5. approaching the center line. If the values are concentrated near the center line, this may indicate that the grouping method was chosen incorrectly, which makes the range too wide and leads to mixing of data from different distributions.

6. frequency. When, after certain equal periods of time, the curve goes either “declining” or “rising”.

VII. Analysis of control charts.

Further actions are based on the conclusion about the stability or instability of the process. If the process does not meet the stability criteria, the influence of non-random factors should be reduced and a control chart should be built by collecting new data. But, if the process meets the stability criteria, it is necessary to evaluate the process capabilities (Cp). The smaller the spread of parameters within the tolerance limits, the higher the value of the process capability indicator. The indicator reflects the ratio of the width of the parameter and the degree of its scatter. The opportunity index is calculated as

, where can be calculated as .

If the calculated indicator is less than 1, then the researcher needs to improve the process, either stop production of the product, or change the requirements for the product. With index value:<1 возможности процесса неприемлемы,

Wed

Cр>1 process satisfies the criterion of possibility.

In the case of no displacement relative to the center line Cp=Cpk, where . These two indicators are always used together to determine the status of the process, so in mechanical engineering it is considered the norm , which means that the probability of non-compliance does not exceed 0.00006.

Now, having considered the algorithm for constructing control charts, let’s look at a specific example.

Task: The chromium content in steel castings is controlled. Measurements are taken in four swimming trunks. Table 2 shows data for 15 subgroups. It is necessary to build a map.

Solution: Since we already know in advance what type of map needs to be built, let’s calculate the values

subgroup number	X1	X2	X3	X4		R
1	0,74	0,76	0,62	0,73	0,713	0,14
2	0,72	0,74	0,84	0,69	0,748	0,15
3	0,87	0,79	0,70	0,92	0,820	0,22
4	0,78	0,66	0,71	0,74	0,723	0,12
5	0,81	0,66	0,82	0,67	0,740	0,16
6	0,63	0,71	0,68	0,82	0,710	0,19
7	0,63	0,73	0,64	0,80	0,700	0,17
8	0,66	0,68	0,85	0,91	0,775	0,25
9	0,63	0,66	0,62	0,85	0,690	0,23
10	0,85	0,61	0,75	0,77	0,745	0,24
11	0,73	0,65	0,74	0,90	0,755	0,25
12	0,85	0,77	0,65	0,69	0,740	0,20
13	0,67	0,69	0,83	0,62	0,703	0,21
14	0,74	0,73	0,62	0,88	0,743	0,26
15	0,81	0,82	0,69	0,73	0,763	0,13
average:	0,738	0,19

The next step is to calculate , where, in accordance with the above scheme, , and . Now, having the values of the central line, the average value of the indicator and the average deviation, we will find the values of the control boundaries of the cards.

, where is found in the table of coefficients for calculating control chart lines and is equal to 0.729. Then UCL=0.880, LCL=0.596.

For values, the lower and upper control limits are determined by the formulas:

where and are found in the table of coefficients for calculating control chart lines and are equal to 0.000 and 2.282, respectively. Then UCL=0.19*2.282=0.444 and LCL=0.19*0.000=0.

Let's build control charts for the average values and ranges of this sample using Excel:

As far as we can verify, the control charts did not reveal non-random values, departures from control limits, series or trends. However, the graph of average values gravitates towards the central position, which may indicate both incorrectly chosen tolerance limits, an abnormality of the distribution and instability of the process. To make sure, let’s calculate the process capability index. , where can be calculated as , using the table of coefficients, we find a value equal to ;

Since the calculated index<1, что свидетельствует о неприемлемости возможностей процесса, его статистической неуправляемости и не стабильности. Необходимо провести усовершенствования процесса, установить контроль над его протеканием, с целью уменьшения влияния не случайных факторов.

By studying specialized literature and delving into quality management, I was able to glean a large amount of interesting and useful information. For example, the breadth of use of quality management has affected all areas of production from heavy industry and oil production to small organizations providing services (catering places, bookstores, etc.).

In recent years, under the pervasive influence of thinking aimed at improving quality and customer satisfaction, systems such as CRM—customer-oriented management—are attributed to quality management; ERP enterprise resource management system; TPM is a total equipment maintenance system, and many other systems. Based on this, we can conclude that there has been a shift in interests from managing the quality of a specific process to the use of quality systems and software packages that allow one way or another to help meet customer needs in the most convenient ways. Walter Shewhart's contribution to statistical quality management is great, and the control charts he proposed are still used, but more often, together with other methods, due to the provision of a systematic approach and consideration of many factors that were not taken into account back in the 20th century.

In conclusion, I would like to say that the main problem of modern quality systems is that, despite their apparent ease of use, they cannot guarantee their effective use in the enterprise. The reasons lie in the origins! After all, the main advantage of using the “7 simple methods” of quality management is that without penetration by the philosophy of quality, obtaining any significant results is unlikely. Thus, companies that are not yet ready for fundamental changes could protect themselves from introducing expensive systems and unnecessary expenses.

Quality management is the philosophy of success of modern companies!

1. GOST R 50779.42-99 “Statistical methods. Shewhart control charts"

2. Goldratt E.M., Cox J. Purpose. Continuous Improvement Process/E.M. Goldratt, J. Cox. - Potpourri Publishing House - 2007.

3. Yoshio Kondo. Quality management on a company scale: formation and stages of development./ trans. from English E.P. Markova, I.N. Rybakov. - Nizhny Novgorod: SMC “Priority”, 2002.

4. Prosvetov G.I. Forecasting and planning: tasks and solutions: educational manual./G.I. Prosveov-M.: RDL Publishing House, 2005.

5. Kane M.M., Ivanov B.V., Koreshkov V.N., Skhirtladze A.G. Systems, methods and tools of quality management / M.M. Kahne, B.V. Ivanov, V.N. Koreshkov, A.G. Skhirtladze. – St. Petersburg: Peter, 2009

6. Kachalov V.A. What is “continuous improvement of QMS effectiveness”? // Methods of quality management. - 2006. - No. 10.

7. Klyachkin V.N. Statistical methods in quality management: computer technologies: textbook. Manual/V.N. Klyachkin.-M.: Finance and Statistics, 2007.

8. Kruglov M.G., Shishkov G.M. Quality management as it is / M.G. Kruglov, G.M. Shishkov.-M.: Eksmo, 2006.

9. Kuznetsov L.A. Control and assessment of multidimensional quality//methods of quality management.-2008.-No. 10.-P. 40-45.

10. Sazhin Yu.V., Pletneva N.P. On the issue of the effectiveness of QMS in Russia // Methods of quality management. - 2008. - No. 10. - P. 20-24.

11. Statistical methods for improving quality: monograph / trans. from English Y.P.Adler, L.A. Konareva; edited by Kume.-M.: Finance and Statistics, 1990.

12. Feigenbaum A. Product quality control/A. Feigenbaum. - M.: Economics, 1986.

13. Evans J. Quality management: textbook. Allowance/J. Evans.-M.: Unity-Dana, 2007.

Schematic of Shewhart control charts

Coefficients for calculating control chart lines.

Kane M.M., Ivanov B.V., Koreshkov V.N., Skhirtladze A.G. Systems, methods and tools of quality management / M.M. Kahne, B.V. Ivanov, V.N. Koreshkov, A.G. Skhirtladze. – St. Petersburg: Peter, 2009

Kane M.M., Ivanov B.V., Koreshkov V.N., Skhirtladze A.G. Systems, methods and tools of quality management / M.M. Kahne, B.V. Ivanov, V.N. Koreshkov, A.G. Skhirtladze. – St. Petersburg: Peter, 2009.

I recently published my own here, where in quite simple language, in places abusing foul language, to the 20-minute laughter of the listeners, I talked about how to separate systemic variations from variations caused by special reasons.

Now I want to look in detail at an example of constructing a Shewhart control chart based on real data. As real data, I took historical information about completed personal tasks. I have this information thanks to adapting David Allen’s Getting Things model of personal effectiveness (I also have an old slidecast about this in three parts: Part 1, Part 2, Part 3 + Excel spreadsheet with macros for analyzing tasks from Outlook).

The problem statement looks like this. I have a distribution of the average number of completed tasks depending on the day of the week (below in the graph) and need to answer the question: “is there anything special about Mondays or is it just a system error?”

Let's answer this question using the Shewhart control chart - the main tool for statistical process control.

So, Shewhart's criterion for the presence of a special cause of variation is quite simple: if some point goes beyond the control limits calculated in a special way, then it indicates a special cause. If the point lies within these limits, then the deviation is due to the general properties of the system itself. Roughly speaking, it is a measurement error.
The formula for calculating control limits is:

Where
- average value of the average values for the subgroup,
- average range,
- some engineering coefficient depending on the size of the subgroup.

All formulas and tabular coefficients can be found, for example, in GOST 50779.42-99, where the approach to statistical management is briefly and clearly outlined (honestly, I myself did not expect that there was such a GOST. The topic of statistical management and its place in business optimization is covered in more detail in book by D. Wheeler).

In our case, we group the number of completed tasks by day of the week - these will be the subgroups of our sample. I took data on the number of completed tasks over 5 weeks of work, that is, the size of the subgroup is 5. Using Table 2 from GOST, we find the value of the engineering coefficient:

Calculating the average value and range (the difference between the minimum and maximum values) by subgroup (in our case, by day of the week) is a fairly simple task, in my case the results are as follows:

The central line of the control chart will be the average of the group means, that is:

We also calculate the average range:

We now know that the lower control limit for the number of tasks completed will be:

That is, those days on which I complete fewer tasks on average are special from the system's point of view.

Similarly, we obtain the upper control limit:

Now let's plot the center line (red), upper control limit (green) and lower control limit (purple):

And, oh, miracle! We see three clearly special groups outside the control limits, in which there are clearly non-systemic causes of variation!

I don't work on Saturdays and Sundays. Fact. And Monday turned out to be a truly special day. And now you can think and look for what is really special about Mondays.

However, if the average number of tasks completed on Monday was within the control limits and even stood out strongly against the background of other points, then from the point of view of Shewhart and Deming, looking for any peculiarities on Mondays would be a pointless exercise, since such behavior is determined exclusively by general reasons . For example, I built a control chart for another 5 weeks at the end of last year:

And there seems to be some feeling that Monday stands out somehow, but according to the Shewhart criterion, this is just a fluctuation or an error in the system itself. According to Shewhart, in this case, you can study the special causes of Mondays for as long as you like - they simply do not exist. From the point of view of the statistical office, in these data Monday is no different from any other working day (even Sunday).

4. Examples of constructing Shewhart control charts using GOST R 50779.42–99

Shewhart control charts come in two main types: for quantitative and alternative data. For each control chart there are two situations:

a) standard values are not specified;

b) standard values are set.

Standard values are values established in accordance with some specific requirement or purpose.

The purpose of control charts for which no standard values are specified is to detect deviations in the values of characteristics (for example, or some other statistic) that are due to causes other than those that can be explained only by chance. These control charts are based entirely on data from the samples themselves and are used to detect variations that are due to non-random causes.

The purpose of control charts, given given standard values, is to determine whether the observed values differ, etc. for several subgroups (each with a volume of observations) from the corresponding standard values (or), etc. more than can be expected from the action of random causes alone. A special feature of maps with given standard values is the additional requirement related to the position of the center and the variation of the process. Established values may be based on experience gained from the use of control charts at specified standard values, as well as on economics determined after consideration of service needs and production costs, or specified in product specifications.

4.1 Control charts for quantitative data

Quantitative control charts are classic control charts used for process control where the characteristics or results of a process are measurable and the actual values of the controlled parameter measured to the required accuracy are recorded.

Control charts for quantitative data allow you to control both the location of the center (level, mean, center of tuning) of the process and its spread (range, standard deviation). Therefore, control charts for quantitative data are almost always used and analyzed in pairs—one chart for location and the other for scatter.

The most commonly used pairs are and -cards, as well as -cards. Formulas for calculating the position of the control boundaries of these maps are given in Table. 1. The values of the coefficients included in these formulas and depending on the sample size are given in Table. 2.

It should be emphasized that the coefficients given in this table were obtained under the assumption that the quantitative values of the controlled parameter have a normal or close to normal distribution.

Table 1

Control limit formulas for Shewhart charts using quantitative data

Statistics			Standard values are set
Statistics	Central line	UCL and LCL	Central line	UCL and LCL



Note: the default values are either , , or .

table 2

Coefficients for calculating control chart lines

Number of observations in sub-group n	Coefficients for calculating control limits											Coefficients for calculating the center line
Number of observations in sub-group n
2	2,121	1,880	2,659	0,000	3,267	0,000	2,606	0,000	3,686	0,000	3,267	0,7979	1,2533	1,128	0,8865
3	1,732	1,023	1,954	0,000	2,568	0,000	2,276	0,000	4,358	0,000	2,574	0,8886	1,1284	1,693	0,5907
4	1,500	0,729	1,628	0,000	2,266	0,000	2,088	0,000	4,696	0,000	2,282	0,9213	1,0854	2,059	0,4857
5	1,342	0,577	1,427	0,000	2,089	0,000	1,964	0,000	4,918	0,000	2,114	0,9400	1,0638	2,326	0,4299
6	1,225	0,483	1,287	0,030	1,970	0,029	1,874	0,000	5,078	0,000	2,004	0,9515	1,0510	2,534	0,3946
7	1,134	0,419	1,182	0,118	1,882	0,113	1,806	0,204	5,204	0,076	1,924	0,9594	1,0423	2,704	0,3698
8	1,061	0,373	1,099	0,185	1,815	0,179	1,751	0,388	5,306	0,136	1,864	0,9650	1,0363	2,847	0,3512
9	1,000	0,337	1,032	0,239	1,761	0,232	1,707	0,547	5,393	0,184	1,816	0,9693	1,0317	2,970	0,3367
10	0,949	0,308	0,975	0,284	1,716	0,276	1,669	0,687	5,469	0,223	1,777	0,9727	1,0281	3,078	0,3249
11	0,905	0,285	0,927	0,321	1,679	0,313	1,637	0,811	5,535	0,256	1,744	0,9754	1,0252	3,173	0,3152
12	0,866	0,266	0,886	0,354	1,646	0,346	1,610	0,922	5,594	0,283	1,717	0,9776	1,0229	3,258	0,3069
13	0,832	0,249	0,850	0,382	1,618	0,374	1,585	1,025	5,647	0,307	1,693	0,9794	1,0210	3,336	0,2998
14	0,802	0,235	0,817	0,406	1,594	0,399	1,563	1,118	5,696	0,328	1,672	0,9810	1,0194	3,407	0,2935
15	0,775	0,223	0,789	0,428	1,572	0,421	1,544	1,203	5,741	0,347	1,653	0,9823	1,0180	3,472	0,2880
16	0,750	0,212	0,763	0,448	1,552	0,440	1,526	1,282	5,782	0,363	1,637	0,9835	1,0168	3,532	0,2831
17	0,728	0,203	0,739	0,466	1,534	0,458	1,511	1,356	5,820	0,378	1,622	0,9845	1,0157	3,588	0,2784
18	0,707	0,194	0,718	0,482	1,518	0,475	1,496	1,424	5,856	0,391	1,608	0,9854	1,0148	3,640	0,2747
19	0,688	0,187	0,698	0,497	1,503	0,490	1,483	1,487	5,891	0,403	1,597	0,9862	1,0140	3,689	0,2711
20	0,671	0,180	0,680	0,510	1,490	0,504	1,470	1,549	5,921	0,415	1,585	0,9869	1,0133	3,735	0,2677
21	0,655	0,173	0,663	0,523	1,477	0,516	1,459	1,605	5,951	0,425	1,575	0,9876	1,0126	3,778	0,2647
22	0,640	0,167	0,647	0,534	1,466	0,528	1,448	1,659	5,979	0,434	1,566	0,9882	1,0119	3,819	0,2618
23	0,626	0,162	0,633	0,545	1,455	0,539	1,438	1,710	6,006	0,443	1,557	0,9887	1,0114	3,858	0,2592
24	0,612	0,157	0,619	0,555	1,445	0,549	1,429	1,759	6,031	0,451	1,548	0,9892	1,0109	3,895	0,2567
25	0,600	0,153	0,606	0,565	1,434	0,559	1,420	1,806	6,056	0,459	1,541	0,9896	1,0105	3,931	0,2544

An alternative to maps are median control charts (– maps), the construction of which involves less computation than maps. This may make it easier to introduce them into production. The position of the central line on the map is determined by the average value of the medians () for all tested samples. The positions of the upper and lower control limits are determined by the relations

(4.1)

The values of the coefficient depending on the sample size are given in table. 3.

Table 3

Coefficient values

	2	3	4	5	6	7	8	9	10
	1,88	1,19	0,80	0,69	0,55	0,51	0,43	0,41	0,36

Usually - map is used together with - map, sample size

In some cases, the cost or duration of measuring the controlled parameter is so great that it is necessary to control the process based on measuring individual values of the controlled parameter. In this case, the sliding range serves as a measure of process variation, i.e. the absolute value of the difference in measurements of the monitored parameter in successive pairs: the difference between the first and second measurements, then the second and third, etc. Based on the moving ranges, the average moving range is calculated, which is used to construct control charts of individual values and moving ranges (and -maps). Formulas for calculating the position of the control boundaries of these maps are given in Table. 4.

Table 4

Control limit formulas for individual value maps

Statistics	No default values specified		Standard values are set
Statistics	Central line	UCL and LCL	Central line	UCL and LCL
Individual meaning
Sliding
Note: the default values are and or and .

The values of the coefficients and can be indirectly obtained from Table 2 with n=2.

4.1.1 and -cards. No default values specified

In table Figure 6 shows the results of measurements of the outer radius of the bushing. Four measurements were taken every half hour, for a total of 20 samples. The means and ranges of the subgroups are also shown in Table. 5. The maximum permissible values for the outer radius are established: 0.219 and 0.125 dm. The goal is to determine the performance of the process and control it in terms of tuning and variation so that it meets the specified requirements.

Table 5

Manufacturing Data for Bushing Outer Radius

Subgroup number	Radius
Subgroup number
1	0,1898	0,1729	0,2067	0,1898	0,1898	0,038
2	0,2012	0,1913	0,1878	0,1921	0,1931	0,0134
3	0,2217	0,2192	0,2078	0,1980	0,2117	0,0237
4	0,1832	0,1812	0,1963	0,1800	0,1852	0,0163
5	0,1692	0,2263	0,2066	0,2091	0,2033	0,0571
6	0,1621	0,1832	0,1914	0,1783	0,1788	0,0293
7	0,2001	0,1937	0,2169	0,2082	0,2045	0,0242
8	0,2401	0,1825	0,1910	0,2264	0,2100	0,0576
9	0,1996	0,1980	0,2076	0,2023	0,2019	0,0096
10	0,1783	0,1715	0,1829	0,1961	0,1822	0,0246
11	0,2166	0,1748	0,1960	0,1923	0,1949	0,0418
12	0,1924	0,1984	0,2377	0,2003	0,2072	0,0453
13	0,1768	0,1986	0,2241	0,2022	0,2004	0,0473
14	0,1923	0,1876	0,1903	0,1986	0,1922	0,0110
15	0,1924	0,1996	0,2120	0,2160	0,2050	0,0236
16	0,1720	0,1940	0,2116	0,2320	0,2049	0,0600
17	0,1824	0,1790	0,1876	0,1821	0,1828	0,0086
18	0,1812	0,1585	0,1699	0,1680	0,1694	0,0227
19	0,1700	0,1567	0,1694	0,1702	0,1666	0,0135
20	0,1698	0,1664	0,1700	0,1600	0,1655	0,0100

where is the number of subgroups,

The first step: constructing a map and determining the state of the process from it.

center line:

The values of the factors and are taken from the table. 2 for n=4. Since the values in the table. 5 are within the control limits, the map indicates a statistically controlled state. The value can now be used to calculate map control boundaries.

center line: g

The multiplier values are taken from the table. 2 for n=4.

and -maps are shown in Fig. 5. Analysis of the map shows that the last three points are outside the boundaries. This indicates that some special causes of variation may be at work. If limits have been calculated based on previous data, then action must be taken at the point corresponding to the 18th subgroup.

Fig.5. Medium and large maps

At this point in the process, appropriate corrective action should be taken to eliminate the special causes and prevent their recurrence. Work with the maps continues after the revised control boundaries have been established without excluded points that went beyond the old boundaries, i.e. values for samples No. 18, 19 and 20. The values and lines of the control chart are recalculated as follows:

revised value

The revised map has the following parameters:

center line: g

revised –map:

center line:

(since the center line is: , then there is no LCL).

For a stable process with revised control limits, capabilities can be assessed. We calculate the opportunity index:

where is the upper maximum permissible value of the controlled parameter; – lower maximum permissible value of the controlled parameter; – estimated by the average variability within subgroups and expressed as . The value of the constant is taken from Table 2 for n=4.

Rice. 6. Revised and -maps

Since , the process capabilities can be considered acceptable. However, upon closer examination, it can be seen that the process is not set up correctly relative to the tolerance and therefore about 11.8% of units will fall outside the specified upper limit value. Therefore, before setting constant parameters of control charts, one must try to correctly configure the process, while maintaining it in a statistically controlled state.

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